Classifying and constraining local four photon and four graviton S-matrices

TL;DR

This work classifies all kinematically allowed local four-photon and four-graviton S-matrices that are polynomial in momenta across spacetime dimensions, organizing them as the S3-invariant sector of a local module built over Mandelstam polynomials. It constructs both the bare and local modules, counts S-matrices via representation theory of S3, and provides explicit generators and corresponding Lagrangians; Plethystic counting is used as a nontrivial check. A central proposal is the Classical Regge Growth bound, demanding S ∼ s^2 at fixed t, which, together with causality and unitarity considerations, constrains the space of allowed polynomial gravitons and photons: for photons many CRG-allowed structures exist, while for gravitons in D≤6 no polynomial Einstein modifications survive, and in D≥7 a unique six-derivative Lovelock-like term remains. Exchange contributions are analyzed and shown to generically violate CRG in D≤6, reinforcing the conjectured rigidity of gravitational interactions in the classical limit. These results illuminate the landscape of possible classical gravitational S-matrices and provide a structured framework for connecting local Lagrangians, S-matrices, and high-energy growth, with potential implications for AdS/CFT and the uniqueness of Einstein gravity at low-energy scales.

Abstract

We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants $s$, $t$ and $u$. We construct these modules for every value of the spacetime dimension $D$, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by $s^2$ at fixed $t$. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for $D \leq 6$. For $D \geq 7$ there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for $D\leq 6$. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also such violates our conjectured Regge growth bound, at least when $D\leq 6$, even when the exchanged particles have low spin.

Figures (4)

• Figure 1: Parity even index structures of 4-photon scattering
• Figure 2: Parity even index structures of 4-graviton scattering.
• Figure 3: Parity odd index structures of 4-photon and 4-graviton scattering in $D=7$ respectively. The $\widetilde{\varepsilon}$ symbol is denoted by a red dot with $D-3$ valency.
• Figure 4: The left figure shows that there is a single $\mathbb{Z}_2\times \mathbb{Z}_2$ orbit in subsectors $\tt^{\otimes 3}{\mathtt v}, \tt {\mathtt v}^{\otimes 3}, {\mathtt v}^{\otimes 3}{\mathtt s}$. The right figure shows the three distinct $\mathbb{Z}_2\times \mathbb{Z}_2$ orbits in the subsector $\tt^{\otimes 2}{\mathtt v}{\mathtt s}$.